Investigation of the mathematical structure of bidomain models

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K18732
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Meiji University
• Principal Investigator: Matano Hiroshi 明治大学, 研究・知財戦略機構(中野), 特任教授 (40126165)
• Co-Investigator(Kenkyū-buntansha): 奈良 光紀 岩手大学, 理工学部, 准教授 (90512161)
• Project Period (FY): 2017-06-30 – 2023-03-31
• Keywords: バイドメインモデル / 非線形問題 / 進行波 / 安定性 / 定性的理論 / 擬微分方程式 / 数値シミュレーション / フランク図形

Outline of Final Research Achievements

Bidomain models are very important mathematical models in cardiac electrophysiology. However, it is very difficult to analyze bidomain models mathematically, therefore not much was known about the qualitative properties of solutions such as stability. In the present research, we studied the bidomain Allen-Cahn equation and established a general theory on the nonlinear stability of planar waves, and solved the open problem concerning whether or not the maximum principle holds for the bidomain operator. We also studied the sawtooth zigzag patterns that typically appear when a planar front destabilizes and, by a combination of theoretical and numerical methods, we shed light on the mechanism that produces those sawtooth patterns. Furthermore, by using numerical simulations, we studiend the behavior of pulse waves in the bidomain FitzHugh-Nagumo equation and showed that there are different types of behaviors when the flat pulse waves become unstable.

Free Research Field

非線形解析学

Academic Significance and Societal Importance of the Research Achievements

バイドメインモデルは心臓電気生理学で極めて重要であるが，その定性的性質は長らく未解明であった．これに突破口を開いたのが，我々が2016年に発表したバイドメインAllen-Cahn方程式の平面波の線形安定性に関する論文である．今回の研究は，これをさらに発展させて，理論的解析と数値シミュレーションを併用して未知の部分の多いバイドメインモデルの特性にさまざまな角度から光をあてたものである．バイドメインモデルの定性的性質の研究は，最近始まったばかりであり，いまだ解決すべき問題は数多く残っているが，我々が得た知見が，将来的には不整脈等の理解など，医学生理学分野の研究に資する可能性があると考える．